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Chapter 2: Problem 16

In Problems 13-16, make use of the known graph of \(y=\ln x\) to sketch the graphs of the equations. \(y=\ln (x-2)\)

Short Answer

Expert verified
Shift the graph of \( y = \ln x \) 2 units right; starts at (3,0); asymptote at \( x = 2 \).

Step by step solution

01

Understand the Parent Graph

The parent graph is for the function \( y = \ln x \). This graph is defined for all \( x > 0 \) and passes through the point (1, 0) while increasing slowly and never touching the y-axis (asymptote at \( x = 0 \)).
02

Identify the Transformation

The transformation applied to the parent graph \( y = \ln x \) is a horizontal shift. The function \( y = \ln (x-2) \) is obtained by shifting the graph of \( y = \ln x \) to the right by 2 units. This means the graph will now pass through the point (3, 0).
03

Determine Domain of Transformed Graph

The domain of \( y = \ln (x-2) \) is obtained by solving \( x-2 > 0 \). Thus, the domain is \( x > 2 \). This implies the vertical asymptote is now at \( x = 2 \).
04

Sketch the Transformed Graph

Start the graph at \( x = 2 \), which is the vertical asymptote. Plot the key point (3, 0) where the graph crosses the x-axis. Draw the curve such that it moves up to the right and approaches the x-axis, similar to the parent graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm Function
The natural logarithm function, commonly denoted as \( y = \ln x \), is a fundamental concept in mathematics used to describe the logarithm base \( e \) of a number. The number \( e \) is an irrational constant approximately equal to 2.71828. Let's explore some of the key features and characteristics of this function:
  • The natural logarithm function is only defined for positive values of \( x \). This is why the domain of \( y = \ln x \) is \( x > 0 \).
  • As the input values \( x \) increase, the function \( y = \ln x \) also increases but at a gradually slower rate. This is because its derivative, \( \frac{1}{x} \), represents a decreasing function.
  • A critical point on the graph of \( y = \ln x \) is \( (1, 0) \) because \( \ln 1 = 0 \).
  • The graph approaches but never meets the y-axis, resulting in a vertical asymptote at \( x = 0 \).
Essentially, the natural logarithm reveals the power to which \( e \) must be raised to produce a given number. Its unique properties make it an essential tool in various fields such as calculus and complex analysis.
Horizontal Shifting
Horizontal shifting is a type of transformation applied to the original graph of a function to move it left or right on the coordinate plane. For the function \( y = \ln(x - 2) \), the graph of the parent function \( y = \ln x \) is shifted horizontally. Here's how it works:
  • The expression \( x - 2 \) within the natural logarithm function indicates that the entire graph shifts to the right by 2 units.
  • This shift does not change the shape of the graph, but it alters both the domain and the position of the graph's key features, such as the intercepts and asymptotes.
  • The point where the graph crosses the x-axis, initially at \( (1, 0) \) on the parent graph, now moves to \( (3, 0) \) post-shift.
  • The vertical asymptote, previously at \( x = 0 \), is moved to \( x = 2 \).
Horizontal shifts are straightforward and commonly used in graph transformations to modify the location of functions without affecting their fundamental properties.
Domain of Functions
The domain of a function is the set of all possible input values (or \( x \)-values) that the function can accept. Understanding the domain is crucial for effectively working with functions, particularly involving logarithmic operations. For the function \( y = \ln(x - 2) \), determining the domain involves:
  • Identifying constraints within the function that restrict the possible values of \( x \).
  • Since logarithmic functions like \( y = \ln(x - 2) \) require their argument \((x - 2)\) to be positive, solve the inequality \( x - 2 > 0 \).
  • This inequality simplifies to \( x > 2 \), meaning the function is only defined for \( x \)-values greater than 2.
  • The result is a vertical asymptote at \( x = 2 \), where the function approaches infinity as \( x \) approaches this asymptote from the right.
Understanding the domain gives insights into where and how the function behaves, helping in sketching the graph accurately and predicting the behavior of the function across different intervals.

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Most popular questions from this chapter

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \sqrt{|x|} $$

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0}(\sin 2 x) / 4 x $$

Prove that \(\lim _{n \rightarrow \infty}\left(1-\frac{1}{n}\right)^{-n}=e\). Hint: First show that $$ \left(1-\frac{1}{n}\right)^{-n}=\left(1+\frac{1}{n-1}\right)^{n}=\left(1+\frac{1}{n-1}\right)^{n-1}\left(1+\frac{1}{n-1}\right) $$

In Problems 1-10, simplify the given expression. \(\ln e^{\cos x}\)

In Problems 1-10, simplify the given expression. \(2^{2 \log _{2} x}\)
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